The class will cover the basics of forcing, a technique introduced
by Cohen to prove the independence of the Continuum Hypothesis
from the axioms of mathematics.

Recall that two sets $A$ and $B$ have the same cardinality if
there is a bijective function $f \colon A\rightarrow B$. The
Continuum Hypothesis states that every infinite subset of
${\mathbb R}$ has either the same cardinality as ${\mathbb R}$, or
the same cardinality as ${\mathbb N}$; there are no cardinalities
in between.

It turns out that the Continuum Hypothesis is {\em independent},
meaning neither provable nor refutable, from the axioms of
mathematics. That it cannot be refuted was shown by G\"odel
(1940's) and that it cannot be proved was shown by Cohen (1960's).
Cohen's technique has since been used in proofs of many other
independence results. For example Solovay used it to show that the
existence of a non-measurable set of reals cannot be proved
without the axiom of choice.

The class will cover the basics of the forcing technique,
preservation of cardinals in forcing extensions, applications to
cardinal arithmetic (including the independence of the CH),
iterated forcing, Martin's axiom, and applications of Martin's
axiom.

Time and Place: MWF 12-12:50pm, in MS 6118.

Text: Set Theory, an Introduction to Independence Proofs, by Kenneth Kunen.

Grading and assignments: Students will be asked
to solve assigned questions from Kunen's book
and present the solutions in class.
Grading will be based on the presentations.